Because the rate of heat transfer through a gas is a function of the gas pressure, under certain conditions measurements of heat transfer rates from a heated sensing element to the gas can, with appropriate calibration, be used to determine the gas pressure. This principle is used in the well known Pirani gauge (shown in schematic form in FIGS. 1a and 1b), in which heat loss is measured with a Wheatstone bridge network which serves both to heat the sensing element and to measure its resistance.
Referring to FIG. 1a, in a Pirani gauge the pressure sensor consists of a temperature sensitive resistance RS connected as one arm of a Wheatstone bridge. R2 is typically a temperature sensitive resistance designed to have a negligible temperature rise due to the current i2. R3 and R4 are typically fixed resistences. RS and typically R2 are exposed to the vacuum environment whose pressure is to be measured. FIG. 1b illustrates an alternative bridge configuration.
Pirani gauges have been operated with constant current i1 (as shown in U.S. Pat. No. 3,580,081), or with constant voltage across RS. In these methods, an electrical imbalance of the bridge is created which reflects gas pressure. Pirani gauges have also been operated with constant resistance RS (as shown in U.S. Pat. No. 2,938,387). In this mode, the rate at which energy is supplied is varied with changes in gas pressure, so the rate of change in energy supplied reflects changes in gas pressure. Each method of operation has differing advantages and disadvantages, but the following discussion pertains particularly to the constant resistance method and the configuration of FIG. 1a. 
Voltage VB is automatically controlled to maintain the voltage difference between A and C in FIG. 1a at zero volts. When the potential drop from A to C is zero, the bridge is said to be balanced. At bridge balance the following conditions exist:is=i2,  (1) i4=i3,  (2) isRS=i4R4,  (3) and i2R2=i3R3  (4) Dividing Eq. 3 by Eq. 4 and using Eq. 1 and 2 givesRS=βR2  (5)                               where          ⁢                                           ⁢          β                =                  R4          R3                                    (        6        )            Thus, at bridge balance RS is a constant fraction β of R2.
To achieve a steady state condition in RS at any given pressure, Eq. 7 must be satisfied:Electrical power input to RS=Power radiated by RS+Power lost out ends of RS+Power lost to gas by RS  (7) 
A conventional Pirani gauge is calibrated against several known pressures to determine a relationship between unknown pressure, Px, and the power loss to the gas or more conveniently to the bridge voltage. Then, assuming end losses and radiation losses remain constant, the unknown pressure of the gas Px may be directly determined by the power lost to the gas or related to the bridge voltage at bridge balance.
Because Pirani gauges may be designed to have wide range and are relatively simple and inexpensive, there is a long-felt need to be able to use these gauges as a substitute for much higher priced gauges such as capacitance manometers and ionization gauges. However, existing designs leave much to be desired for accurate pressure measurement, especially at lower pressures.
Prior to 1977, the upper pressure limit of Pirani gauges was approximately 20 Torr due to the fact that at higher pressures the thermal conductivity of a gas becomes substantially independent of pressure in macroscopic size devices. One of the present inventors helped develop the CONVECTRON® Gauge produced and sold by the assignee (Granville-Phillips Company of Boulder Colorado) since 1977 which utilizes convection cooling of the sensing element to provide enhanced sensitivity from 20 to 1,000 Torr. Hundreds of thousands of CONVECTRONO Gauges have been sold worldwide. Recently several imitations have appeared on the market.
Although the CONVECTRON® Gauge filled an unsatisfied need, it has several disadvantages. It has by necessity large internal dimensions to provide space for convection. Therefore, it is relatively large. Because convection is gravity dependent, pressure measurements at higher pressures depend on the orientation of the sensor axis. Also, because the pressure range where gas conduction cooling is predominant does not neatly overlap the pressure range where convection cooling occurs, the CONVECTRON® Gauge has limited sensitivity from approximately 20 to 200 Torr.
To help avoid these difficulties, microminiature Pirani sensors have been developed which utilize sensor-to-wall spacings on the order of a few microns rather than the much larger spacings, e.g., 0.5 in., previously used. See for example U.S. Pat. Nos. 4,682,503 to Higashi et al. and 5,347,869 to Shie et al. W. J. Alvesteffer et al., in an article appearing in J. Vac. Sci. Technol. A 13(6), November/December 1995, describe the most recent work on Pirani gauges known to the present inventors. Using such small sensor to wall spacings provides a pressure dependent thermal conductivity even at pressures above atmospheric pressure. Thus, such microscopic sensors have good sensitivity from low pressure to above atmospheric pressure and function in any orientation.
There are a number of problems with previous attempts to develop microminiature gauges. Although microminiature sensors provide good sensitivity over a large pressure range independent of orientation, their design is extremely complex and fabrication requires numerous elaborate processing steps in highly specialized equipment costing hundreds of thousands of dollars.
Microminiature sensors suffer from the same type of ambient temperature-caused errors as do macroscopic sensors. All of the heat loss terms in Eq. 7 are dependent on ambient temperature and on sensing element temperature at any given pressure. Thus, any attempt at pressure measurement with a Pirani gauge without temperature correction will be confused by non-pressure dependent power losses caused by changes in ambient temperature. All modern Pirani gauges attempt to correct for the errors caused by ambient temperature changes. A widely used means for correcting for such errors is to use for R2 a temperature sensitive compensating element RC in series with a fixed resistance R, as shown in FIGS. 1a and 1b. 
British Patent GB 2105047A discloses the provision of an additional resistor to provide a potential divider. J. H. Leck, at page 58 of Pressure Measurement in Vacuum, Chapman and Hall: London (1964) notes that Hale in 1911 made R2 of the same material and physical dimensions as RS in his Pirani gauge. R2 was sealed off in its own vacuum environment and placed in close proximity to RS. When the pressures at R2 and RS were equal, excellent temperature compensation was achieved. However, at other pressures this means of temperature compensation is not very effective.
To avoid the extra cost and complexity of evacuating and sealing off R2 in a separate bulb, R2 is conventionally placed in the same vacuum environment as RS. By making R2 with a relatively large thermal mass and large thermal losses, self heating of R2 can be made negligible. Leck recommends that R2 be “made in two sections, for example, one of copper and the other Nichrome wire . . . so that the overall temperature coefficient (of R2) just matches that of the Pirani element itself (RS).” According to Leck, this method of temperature compensation has been used by Edwards High Vacuum of Great Britain in the METROVAC® brand gauge. A similar temperature compensation arrangement is used in the CONVECTRON® brand gauge.
However, this technique (using two or more materials in R2 having different temperature coefficients of resistance to approximate the temperature coefficient of RS) is effective only over a narrow range of pressure. In fact the compensation can be made exact only at one, or at most several temperatures as noted in U.S. Pat. No. 4,541,286, which discloses this form of temperature compensation in a Pirani gauge. Also, the inventors have found that configurations with a large thermal mass significantly increase the response time of the gauge to sudden changes in ambient temperature.
The inventors have also found, through extensive computer simulation, that using equal temperature coefficients for RS and R2 as recommended by Leck and as practiced in the prior art does not provide an entirely accurate temperature compensation. The inventors have also found that at pressures less than approximately 5×10−3 Torr, the end losses exceed all other losses combined. The relative loss components as determined by this research (radiation loss, end loss and gas loss components of total loss) are shown in the graph of FIG. 2. At 1×10−5 Torr, the end losses are over 1000 times greater than the gas loss and radiation losses are approximately 100 times greater than the gas loss.
Therefore, temperature change effects in prior art Pirani Gauges are especially troublesome at very low pressures where gas conduction losses are very low. Prior art heat loss gauges cannot measure very low pressures accurately, for example, 1×10−5 Torr. The inventors have discovered that this limitation is a result of failure to maintain end losses in the sensing element sufficiently constant when ambient temperature changes. The Alvesteffer-type Pirani gauge has the capability of indicating pressure in the 10−5 Torr range, but does not provide an accurate indication within that range. For example, if the end losses are not held constant to one part in 5,000 in a typical Pirani gauge, a pressure indication at 1×10−5 Torr may be off by 50% to 100%.
The following analysis shows why prior art designs are ill-suited to correct adequately for ambient temperature changes at low pressures. For convenience in examining the prior art, the problems are explained using examples of gauges with relatively large spacing of sensor element to wall. It should be understood that the same type of problems exist in the much more complex geometries of microminiature gauges, with sensing element-to-wall spacings on the order of a few microns.
FIG. 3 is a schematic representation of a portion 302 of a conventional Pirani gauge using a small diameter wire sensing element 304 and a compensating element 303. Those familiar with Pirani gauge design will appreciate that the components in FIG. 3 are not shown to scale, for ease of explanation and understanding. Typically, small diameter wire sensing element 304 is electrically and thermally joined to much larger electrical connectors 306, 307 which are thermally joined to much larger support structures 308, 309. Let TAL represent the temperature in support structure 308 at the left end of sensing element 304 and TAR represent the temperature in support structure 309 at the right end at any given time t. Let TSL and TSR represent the temperatures at left sensing element connector 306 and right sensing element connector 307 respectively. Let TCL and TCR represent the temperatures at left compensating element connector 310 and right compensating element connector 311 respectively. Let TXL and TXR represent the temperatures a distance ΔX from connectors 306 and 307 respectively. In prior art designs, it has apparently been assumed that all of these temperatures are the same. However, the inventors have found that even seemingly negligible differences assume great importance for low pressure accuracy.
To better understand temperature compensation requirements, it is important to note several facts.                (1) At low pressures, the temperature of RC is determined predominantly by heat exchange between the compensating element connections and the compensating element. This is because at ambient temperature and low pressures, radiation and gas conduction are very inefficient means of exchanging heat from the compensating element to its surroundings relative to heat conduction through the ends of the compensating element. Thus, at low pressures the compensating element temperature will be very close to the average of the temperatures of the connectors at each end of the compensating element as shown in Eq. 8.                               T          AVG                =                                            T              CL                        +                          T              CR                                2                                    (        8        )                    (2) The temperature of the electrically heated sensing element varies from the ends to center, increasing with distance from the cooler supports. Using finite element analysis the inventors have simulated the temperature distribution along the sensing element. It has been found that with equal temperature coefficients of resistance for RS and RC, the temperature Tn of any segment n of the sensing element changes with changes in average temperature TAVG of the compensating element RC at constant pressure at bridge balance so as to maintain a constant difference ΔTn=Tn−TAVG. The difference ΔTn is a function of β and R where R=R2−RC.        (3) According to Eq. 5, the sensing element resistance RS at bridge balance will be maintained at a resistance β times the resistance element R2. As the ambient temperature increases, the compensating element connectors also increase in temperature and thus the temperature and resistance of RC will increase according to Eq. 8. Any increase in the temperature and therefore the resistance of RC causes an increase in the temperature and resistance of all segments of RS at bridge balance.        (4) The power losses out the ends of the sensing element depend on the temperature gradient γ at the ends of the sensing element according to Eq. 9:Power lost out end=k γ  (9) where k is a constant and                               γ          L                =                                                            T                XL                            -                              T                SL                                                    Δ              ⁢                                                           ⁢              X                                ⁢                                           ⁢          at          ⁢                                           ⁢          left          ⁢                                           ⁢          end                                    (        10        )                                          γ          R                =                                                            T                XR                            -                              T                SR                                                    Δ              ⁢                                                           ⁢              X                                ⁢                                           ⁢          at          ⁢                                           ⁢          right          ⁢                                           ⁢                      end            .                                              (        11        )            If γL and or γR vary for any reason, then the end losses will change and the pressure indication will be erroneous.        
To understand in detail a significant deficiency in the prior art of temperature compensation at low pressures, assume that from a steady state, TAR is increased slightly for example by changes in the local ambient temperature environment of the right support structure. Assume TAL remains unchanged. Because TAL is assumed not to change, TCL and TSL will remain unchanged. However, the increase in TAR will cause TCR to increase by conduction of heat through the connection. Thus,       T    AVG    =                    T        CL            +              T        CR              2  will increase. An increase in TAVG will cause an increase in TXL and TXR at bridge balance, which will produce changes in γL and γR. These changes in γL and γR will change the end loss term in Eq. 7, causing an error in pressure measurement dependent on the size of the changes in γL and γR.
The inventors have determined that unless TAL changes in substantially the same way as TAR, sensing element end losses will not remain unchanged whenever ambient temperature changes. Prior art Pirani gauges have not been specifically designed to maintain TAL=TAR to the degree necessary for accurate low pressure measurement.
To understand another important deficiency in prior art temperature compensation, assume that from a steady state, ambient temperature is increased and that ambient temperature conditions are such that TAL=TAR. Further assume the sensing element connectors are of equal length but that the right compensating element connector is substantially longer than the left compensating element connector as is the case in a popular prior art Pirani gauge. Thus, TSL=TSR but TCR will lag behind TCL because of the assumed differences in length. During this lag time when TCL≠TCR, TAVG will change, thus changing TXL and TXR at bridge balance. Thus, γL and γR will continually change during the lag time, producing errors in low pressure indication.
The inventors have determined that unless the sensing element and compensating element connectors have substantially identical physical dimensions and substantially identical thermal properties, sensing element end losses will not remain unchanged when ambient temperature changes. Prior art Pirani gauges have not been specifically designed so that sensing and compensating element connectors have identical physical dimensions and thermal properties.
Another significant deficiency arises (as the inventors have discovered) from differences in mass between the compensating element and the sensing element. Assume that the mass of the compensating element is substantially larger than that of the sensing element as is typically the case. With prior art Pirani gauges it is common practice to make the compensating element large relative to the sensing element and to provide a relatively large heat loss path to the compensating element surroundings so that the heat arising from dissipation of electrical power in RC can be readily dispersed. From a steady state, assume that ambient temperature increases and that TAL=TAR at all times. Thus, it will take a longer time for the compensating element to reach a new steady state temperature relative to the time it will take TSL and TSR to reach a new steady state temperature. During this time (which has been observed to be of several hours duration in a popular prior art Pirani gauge) TAVG will continually change, thus continually changing TXL and TXR at bridge balance. Thus, γL and γR will change during the lag time, sensing element end losses will not remain constant, and errors will be produced in low pressure measurement.
The same type of problems occur if the compensating element is designed to change temperature at a different rate than does the sensing element with change in ambient temperature at bridge balance. Prior art designs such as the Alvesteffer-type device have this deficiency.
From their research, the inventors have determined that, unless the compensating element has been designed to change temperature at the same rate as the sensing element, sensing element end losses continue to change long after ambient temperature has stabilized at a new value. Yet, prior art Pirani gauges have not been designed to meet this requirement.
It has long been known to use for R2 a compensating element RC, with substantially the same temperature coefficient of resistance as the sensing element, in series with a temperature insensitive resistance element R so as to provide temperature compensation for gas losses and end losses which vary as the temperature difference between the sensing element and its surroundings. This method of temperature compensation has been employed in the CONVECTRON® Gauge for many years and is also used in the Alvesteffer gauge.
This method of temperature compensation assumes that if (1) the temperature coefficients of resistance of the sensing and compensating elements are equal; and (2) the change in sensing element resistance can be made to rise in tandem with change in compensating element resistance, then (3) the temperature of the sensing element will rise in tandem with ambient temperature changes. Satisfying these two assumptions is highly desirable, of course, because doing so would assure that the temperature difference between the heated sensing element and the surrounding wall at ambient temperature would remain constant as ambient temperature changes.
However, the inventors have found that prior art gauges which utilize a constant resistance R in series with a temperature sensitive resistance RC for R2 provide only partial temperature compensation as will now be explained.
Assume that in FIG. 1a, R2 is composed of a temperature sensitive compensating element RC and a temperature insensitive resistance R so thatR2=RC+R  (12) Thus, Eq. 5 derived above for bridge balance may be written asRS=β(RC+R)  (13) where β is defined by Eq. 6 above.Further, assume when the ambient temperature environment of the gauge is equal to T1 that the sensing element operates at temperature TS1 and the compensating element operates at temperature TC1. Thus whenTAMBIENT=T1  (14) Eq. 13 may be written asRS(T1)(1+αS(TS1−T1))=β[RC(T1)(1+αC(TC1−T1))+R]  (15) 
Here, RS(T1) is the resistance of the sensing element at temperature T1, αS is the temperature coefficient of resistance of RS at T1, RC(T1) is the resistance of the compensating element at temperature T1, and αC is the temperature coefficient of resistance of RC at T1. Thus, when
 TAMBIENT=T2 
Eq. 13 may be written asRS(T1)(1+αS(TS2−T1))=β[RC (T1)(1+αC(TC2−T1))+R]  (16) Solving Eq. 15 for TS1 gives                               T          S1                =                                                            [                                                                            β                                              RS                        ⁡                                                  (                                                      T                            1                                                    )                                                                                      ⁡                                          [                                                                                                    RC                            ⁡                                                          (                                                              T                                1                                                            )                                                                                ⁢                                                      (                                                          1                              +                                                              α                                ⁢                                                                                                                                   ⁢                                                                  c                                  ⁡                                                                      (                                                                                                                  T                                        C1                                                                            -                                                                              T                                        1                                                                                                              )                                                                                                                                                        )                                                                          +                        R                                            ]                                                        -                  1                                ]                            /              α                        ⁢                                                   ⁢            s                    +                      T            1                                              (        17        )            Solving Eq. 16 for TS2 gives                               T          S2                =                                                            [                                                                            β                                              RS                        ⁡                                                  (                                                      T                            1                                                    )                                                                                      ⁡                                          [                                                                                                    RC                            ⁡                                                          (                                                              T                                1                                                            )                                                                                ⁢                                                      (                                                          1                              +                                                              α                                ⁢                                                                                                                                   ⁢                                                                  c                                  ⁡                                                                      (                                                                                                                  T                                        C2                                                                            -                                                                              T                                        1                                                                                                              )                                                                                                                                                        )                                                                          +                        R                                            ]                                                        -                  1                                ]                            /              α                        ⁢                                                   ⁢            s                    +                      T            1                                              (        18        )            Subtracting Eq. 17 from Eq. 18 gives the temperature change ΔT in the sensing element RS when ambient temperature changes from T1 to T2. Thus,                               Δ          ⁢                                           ⁢          T                =                                            T              S2                        -                          T              S1                                =                                    β              ⁡                              (                                                      RC                    ⁡                                          (                                              T                        1                                            )                                                                            RS                    ⁡                                          (                                              T                        1                                            )                                                                      )                                      ⁢                          (                                                α                  ⁢                                                                           ⁢                  c                                                  α                  ⁢                                                                           ⁢                  s                                            )                        ⁢                          (                                                T                  C2                                -                                  T                  C1                                            )                                                          (        19        )            Note that an effective compensating element is designed so that its temperature closely follows ambient temperature. Thus, to a very good approximation,TC2−T2=TC1−T1 orTC2−TC1=T2−T1  (20) Thus, Eq. 19 may be written as                               Δ          ⁢                                           ⁢          T                =                              β            ⁡                          (                                                RC                  ⁡                                      (                                          T                      1                                        )                                                                    RS                  ⁡                                      (                                          T                      1                                        )                                                              )                                ⁢                      (                                          α                ⁢                                                                   ⁢                c                                            α                ⁢                                                                   ⁢                s                                      )                    ⁢                      (                                          T                2                            -                              T                1                                      )                                              (        21        )            It is evident from Eq. 21 that the temperature change ΔT in the sensing element RS will be equal to the change in ambient temperature T2−T1 only if                                           β            ⁡                          [                                                RC                  ⁡                                      (                                          T                      1                                        )                                                                    RS                  ⁡                                      (                                          T                      1                                        )                                                              ]                                ⁡                      [                                          α                ⁢                                                                   ⁢                c                                            α                ⁢                                                                   ⁢                s                                      ]                          =        1                            (        22        )            Prior art gauges using a temperature sensitive compensating element RC in series with a fixed resistance R for R2 in FIG. 1a provide only partial temperature compensation depending on the choice of β. Commercially available gauges having the design described by Alvesteffer et al., the most recent work on Pirani gauges known to the present inventors, would not satisfy Eq. 22.
As a third problem with prior art gauge designs, the inventors have found that the level of power dissipation in R2 adversely affects accuracy. Prior art Pirani gauges, when configured as in FIG. 1a, have the same pressure dependent current in RS as is in the compensating element at bridge balance. When configured as in FIG. 1b, at balance the same pressure dependent voltage is applied across R2 as across RS. Of course, a pressure dependent current in R2 will cause the temperature of RC to rise above ambient temperature by an amount which varies with pressure.
Prior art Pirani gauges typically use a compensating element of much larger physical dimensions than the sensing element, to dissipate the heat and thus prevent excessive temperature in the compensating element. As noted above, different physical dimensions for the sensing and compensating elements cause measurement errors when ambient temperature changes.
A fourth problem is that prior art Pirani gauges produce shifts in pressure indications at low pressures when ambient temperature changes. Prior art Pirani gauges have used a variety of components in attempting to maintain the power lost by the sensing element unchanged as ambient temperature changes. For example, in U.S. Pat. No. 4,682,503 thermoelectric cooling is used to control ambient temperature and thus minimize ambient temperature changes.
In the device disclosed in U.S. Pat. No. 4,541,286, a thermally sensitive element is mounted adjacent to the compensating arm of the bridge (actually glued to the exterior of the vacuum enclosure in a commercial version). Alvesteffer et al. use an additional element (designated therein as R4) in the bridge to help compensate for the fact that the temperature coefficient of resistance is slightly different for the sensing element at operating temperature, compared to the compensating element at ambient temperature. Although each of these prior art hardware fixes remove some of the errors caused by changes in ambient temperature, none of them removes substantially all of the errors. Thus prior art Pirani gauges produce significant shifts in pressure indications at low pressures when ambient temperature changes.
Another prior system, disclosed in U.S. Pat. No. 5,608,168, links various electrical measurements of the bridge (or approximations thereof) and determines the value or temperature of the temperature dependent resistance, and takes this parameter into account in determining the pressure measurement. However, this system has increased complexity because of the need to measure temperatures or other values.
Thus, there is a need for an improved Pirani-type gauge which overcomes these problems.